A Formal Algebra for OLAP

TL;DR

This paper introduces a formal algebra for OLAP operations, providing clear semantics and proofs of correctness, which can serve as a foundation for developing more reliable data analysis tools.

Contribution

It presents the first formal proof of correctness for OLAP operations, defining a precise algebra with well-founded semantics for complex multidimensional data manipulation.

Findings

Defined a set of atomic OLAP transformations with clear semantics

Proved that these transformations can be composed to express complex queries

Established a formal, correct algebra for OLAP operations

Abstract

Online Analytical Processing (OLAP) comprises tools and algorithms that allow querying multidimensional databases. It is based on the multidimensional model, where data can be seen as a cube, where each cell contains one or more measures can be aggregated along dimensions. Despite the extensive corpus of work in the field, a standard language for OLAP is still needed, since there is no well-defined, accepted semantics, for many of the usual OLAP operations. In this paper, we address this problem, and present a set of operations for manipulating a data cube. We clearly define the semantics of these operations, and prove that they can be composed, yielding a language powerful enough to express complex OLAP queries. We express these operations as a sequence of atomic transformations over a fixed multidimensional matrix, whose cells contain a sequence of measures. Each atomic transformation produces a new measure. When a sequence of transformations defines an OLAP operation, a flag is produced indicating which cells must be considered as input for the next operation. In this way, an elegant algebra is defined. Our main contribution, with respect to other similar efforts in the field is that, for the first time, a formal proof of the correctness of the operations is given, thus providing a clear semantics for them. We believe the present work will serve as a basis to build more solid practical tools for data analysis.